If f(x) = ax² + bx + c and f(0) = 2, f(1) = 1 and f(4) = 6, then let us calculate the values of a, b and c.

We have,

f(x) = ax² + bx + c …(i)

f(0) = 2

f(1) = 1

f(4) = 6

Replace x by 0 in equation (i), we get

f(0) = a(0)² + b(0) + c

⇒ 2 = 0 + 0 + c [∵ f(0) = 2]

⇒ 2 = c

⇒ c = 2 …(ii)

Now, replace x by 1 in equation (i), we get

f(1) = a(1)² + b(1) + c

⇒ 1 = a + b + 2 [∵ f(1) = 1 & c = 2]

⇒ a + b = 1 – 2

⇒ a + b = -1 …(iii)

Finally, replace x by 4 in equation (i), we get

f(4) = a(4)² + b(4) + c

⇒ 6 = 16a + 4b + 2 [∵ f(4) = 6 & c = 2]

⇒ 16a + 4b = 6 – 2

⇒ 16a + 4b = 4

⇒ 4 (4a + b) = 4

⇒ 4a + b = 1 …(iv)

Solving equations (iii) & (iv), we get

⇒ -3a = -2

⇒

Put a = 2/3 in equation (iii), we get

⇒

⇒

⇒

Thus, a = 2/3, b = -5/3 and c = 2.